Rigid Objects in Equilibrium &amp; Center of Gravity

A lunch tray is being held in one hand, as the drawing illustrates. The mass of the tray itself is 0.220 kg, and its center of gravity is located at its geometrical center. On the tray is a 1.00 kg plate of food and a 0.350 kg cup of coffee. Obtain the force T exerted by the thumb and the force F exerted by the four fingers. Both forces act perpendicular to the tray, which is being held parallel to the ground.

For some reason, you changed your pivot point to be the left side of the tray. In that case: (1) T also provides a clockwise torque; (2) You need to include the torque from F (which would be counterclockwise).

Combine this torque equation (once you correct it) with an equation for the total force on the tray. Solve them together to find T and F.

Your original choice of using the point of application of force F as your pivot point was a good one, since it eliminates force F and allows you to immediately calculate T. (Your only error was using an incorrect distance for one of the forces.) You'd still need another equation to find F.